#246453

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59056	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1322 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1322 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,12)(8,16)(9,11)(10,14)(13,15)(17,19)(18,20)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(1,2)(3,4)(6,11)(7,9)(12,14)(13,16)(15,18)(19,20)(24,33)(27,32)(29,36)(31,34);(2,4)(5,7,10,9)(6,12,14,11)(8,15,17,18)(13,19,20,16)(21,23,25,30)(22,26,28,35)(24,34,33,31)(27,36,32,29);(5,8,10,17)(6,13,14,20)(7,15,9,18)(11,16,12,19)(24,33)(27,32)(29,36)(31,34)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153649);